We derive the conditions under which time-varying cointegration leads to cointegration spaces that may be time-invariant or, in contrast, time-varying. The model of interest is a vector error correction model with arbitrary time-varying cointegration parameters. We clarify the role of identification and normalization restrictions and show that structural breaks in error-correction models may actually correspond to stable long-run economic relationships, as opposed to a single-equation setup, in which an identification restriction is imposed. Moreover, we show that, in a time-varying cointegrating relationship with a given number of variables and cointegration rank, there is a minimum number of orthogonal Fourier functions that most likely guarantees time-varying cointegrating spaces.